The probability of occurrence of events in each of the independent tests is 0.25.

The probability of occurrence of events in each of the independent tests is 0.25. Find the probability that the event will occur 50 times in 243 challenges.

The number of tests is quite large, so we use the local Moivre-Laplace formula:
Pn (k) = φ (x) / √ (n * p * q), where
x = (k – n * p) / √ (n * p * q);
p = 0.25;
q = 1 – p = 0.75;
k = 50;
n = 243.

x = (50 – 243 * 0.25) / √ (243 * 0.25 * 0.75) = – 1.59.

We find the value of φ (x) from the table. The function is even, therefore: φ (x) = φ (- x)
φ (x) = φ (- 1.59) = φ (1.59) = 0.1127.

P243 (50) = 0.1127 / √ (243 * 0.25 * 0.75) = 0.0167.

Answer: The probability of the event occurring 50 times in 243 challenges is 0.0167.